Pion-photon transition distribution amplitudes in the Nambu –Jona-Lasinio model
A. Courtoy*and S. Noguera†Departamento de Fisica Teorica and Instituto de Fı´sica Corpuscular, Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain
(Received 30 July 2007; published 28 November 2007)
We define the pion-photon transition distribution amplitudes (TDA) in a field theoretic formalism from a covariant Bethe-Salpeter approach for the determination of the bound state. We apply our formalism to the Nambu–Jona-Lasinio model, as a realistic theory of the pion. The obtained vector and axial TDAs satisfy all features required by general considerations. In particular, sum rules and the polynomiality condition are explicitly verified. We have numerically proved that the odd coefficients in the polyno-miality expansion of the vector TDA vanish in the chiral limit. The role of PCAC and the presence of a pion pole are explicitly shown.
DOI:10.1103/PhysRevD.76.094026 PACS numbers: 13.60.r, 11.10.St, 12.38.Lg, 24.10.Jv
I. INTRODUCTION
Hard reactions provide important information for un-veiling the structure of hadrons. The large virtuality Q2 involved in the processes allows the factorization of the hard (perturbative) and soft (nonperturbative) contribu-tions in their amplitudes. Therefore these reaccontribu-tions are receiving great attention by the hadronic physics commun-ity. In the past, only total cross sections of inclusive pro-cesses or longitudinal asymmetries, that have simple parton model interpretations, were studied. The basic theo-retical ingredients to be understood are diagonal parton distribution functions (PDF) [1], governing the deep in-clusive processes. In recent years a new variety of pro-cesses, like the deeply virtual Compton scattering, has been considered. These processes are governed by the generalized parton distributions (GPD) [2–5]. From a theoretical point of view, PDFs are related to diagonal matrix elements of a bilocal operator from the initial hadron state to the same final hadron state (the same particle with the same momentum). The GPDs are related to the matrix elements of the same bilocal operator as in the previous case, where the initial and final hadron are the same particle, but have different momentum. The GPDs describe nonforward matrix elements of light-cone opera-tors and therefore measure the response of the internal structure of the hadrons to the probes.
The generalization of parton distributions to the case where the initial and final states correspond to different particles has recently been proposed in [6,7]. Such distri-butions are called transition distribution amplitudes (TDA) since they have been introduced through hadron-photon transitions. In particular, the easiest case is to consider pion-photon TDA, governing processes like !
or ! in the kinematical regime where the virtual photon is highly virtual but with small momen-tum transfer.
The aim of this work is to calculate the pion-photon TDAs in a field theoretic scheme treating the pion as a bound state in a fully covariant manner using the Bethe-Salpeter equation. In this way we preserve all invariances of the problem. In order to perform a numerical study we will use the Nambu–Jona-Lasinio (NJL) model to describe the pion structure. The NJL model is the most realistic model for the pion based on a local quantum field theory built with quarks. It respects the realizations of chiral symmetry and gives a good description of the low energy physics of the pion [8].
The NJL model is a nonrenormalizable field theory and therefore a cutoff procedure has to be defined. We have chosen the Pauli-Villars regularization procedure because it respects all the symmetries of the problem. The NJL model together with its regularization procedure is re-garded as an effective theory of QCD. Moreover, it has been used to tune coefficients of chiral perturbation theory [9].
The NJL model has been used to describe the soft (non-perturbative) part of the deep processes, while for the hard part conventional perturbative QCD must be used. It has been applied to the study of pion PDF [10–12] and to the pion GPD [13]. In the chiral limit, its quark valence dis-tribution is as simple as qx x1 x. Once evolu-tion is taken into account, a good agreement is reached between the calculated PDF and the experimental one [10]. A more elaborate study of pion PDF is done in Ref. [14] using nonlocal Lagrangians [15], which confirms that the result obtained in the NJL model for the PDF is a good approximation.
As GPDs, TDAs must satisfy sum rules. The vector and axial TDA are connected to the vector and axial form factors, FV and FA, appearing in the ! e
process. In order to have a proper understanding of the axial TDA we will clarify the difference between two contributions to the axial current coming from the analysis of the amplitude for the pion radiative decay. The first one is originated from the internal structure of the hadron, in our case a pion. This contribution is the proper axial TDA. A second
contribu-*Aurore.Courtoy@uv.es
tion is present and it can be understood as a manifestation of PCAC. The axial current can be coupled to a pion. This pion is a virtual one and its contribution will be present independent of the external hadron.
The polynomiality condition is also satisfied. We ob-serve that, in the polynomial expansion of the moments of the vector TDA, the odd powers of are chirally sup-pressed and that they vanish in the chiral limit.
Previous studies of the axial and vector pion-photon TDA have been done using different quark models [16,17]. Both works parametrize the TDAs by means of double distributions.
This paper is organized as follows. In Sec. II we estab-lish the connection between the TDA and the vector and axial pion form factors, FV and FA. In Sec. III we define our approach for the TDA and we calculate them in the NJL model. In Sec. IV we study the sum rules and the polynomiality condition of the TDAs. In Sec. V we discuss our results and we finally give our conclusions in Sec. VI.
II. PION-PHOTON TRANSITION DISTRIBUTION AMPLITUDES
The pion-photon TDAs are connected, through sum rules, to the vector and axial-vector pion form factors, FV
and FA. Before giving a proper definition of the TDAs let
us recall the definition of these form factors. They appear in the vector and axial-vector hadronic currents contribut-ing to the decay amplitude of the process ! e
. The precise definitions of these currents are [18,19]
hp0j q0 q0jpi ie" p0 p FVt m ; (1) hp0j q0 5q0jpi e"p0 p gp0 p FAt m e"p0 p p 2p2f m2 t p2fg ; (2)
with f 93 MeV, "0123 1, and
1 i2=2. All the structure of the decaying pion is included in the form factors FVand FA. We observe that the vector current only contains a Lorentz structure associated with the FV form factor. The axial current is composed of two terms. The first one, defining FA, gives the structure of the pion.
The second one corresponds to the axial current for a pointlike pion. It has two different contributions. The first one corresponds to a pointlike coupling between the in-coming pion, the outin-coming photon, and a virtual pion which is coupled to the axial current. It is depicted in the diagram of Fig. 1 and can be seen as a result of PCAC
because the axial current must be coupled to the pion. It isolates the pion pole contribution of the axial current in a model independent way. The second contribution of this term, proportional to fg, corresponds to a pion-photon-axial current contact term. With these definitions, all the structure of the pion remains in the form factor FA.
Let us go now to TDAs. For their definition we introduce the light-cone coordinates v v0 v3=p2 and the transverse components ~v? v1; v2 for any four-vector
v. We define P p p0=2 and the momentum trans-fer, p0 p, therefore P2 m2
=2 t=4 and t 2.
The skewness variable describes the loss of plus momen-tum of the incident pion, i.e., p p0
=2P, and its value ranges between t=2m2
t < < 1. Actually there
is no symmetry relating the distributions for negative and positive which could have constrained the values of the skewness variable to be positive, like for GPDs. The vector and axial TDAs are the Fourier transform of the matrix element of the bilocal currents, qz=2 5qz=2, separated by a lightlike distance. Then, they are directly related to the currents defined in Eqs. (1) and (2) through the sum rules:
Z1 1 dxZ dz 2 e ixPzhp0j qz 2 5q z 2 jpijzz?0 1 Php 0j q0 5q0jpi: (3)
With this connection we can introduce the leading twist decomposition of the bilocal currents. For that we intro-duce the light-front vectors p P1; 0; 0; 1=p2 and
n 1; 0; 0; 1=p2P. The explicit expressions for the pion and photon momenta in terms of their light-cone components are given by Eqs. (A1) and (A2). Then we have Z dz 2 e ixPz hp0j qz 2 n 6 q z 2 jpijzz?0 i Pe" nP V x; ; t 2 p f ; (4)
FIG. 1. Pion pole contribution between the axial current (rep-resented by a cross) and the photon-external pion vertex asso-ciated to the last contribution of Eq. (2).
Z dz 2 e ixPz hp0j qz 2 n 6 5q z 2 jpijzz?0 1 Pe ~" ? ~?A x; ; t 2 p f 1 Pe" 2p2f m2 t x 2 ; (5)
where is equal to 1 for > 0, and equal to 1 for < 0. Here Vx; ; t and Ax; ; t are, respectively, the vector and axial TDAs. They are defined as dimensionless quan-tities. From the condition (3) we observe that they obey the following sum rules:
Z1 1 dxVx; ; t 2 p f m FVt; (6) Z1 1 dxAx; ; t 2 p f m FAt; (7)
which were first introduced in Refs. [6,7].
In the second term of Eq. (5), we have introduced the pion distribution amplitude (PDA) x. By definition the PDA is Z dz 2 e ix1=2pz h0j q z 2 n 6 5q z 2 jpijzz?0 1 pi 2 p f x: (8)
The PDA vanishes outside the region x 2 0; 1 and sat-isfies the normalization condition
Z1 0
dx x 1: (9)
This second term has been introduced in order to isolate the pion pole contribution of the axial current in a model independent way, as we have done in Eq. (2) for the !
e process. Therefore, all the structure of the pion remains in the TDA Ax; ; t. It can be seen as a result of PCAC because the axial current must be coupled to the pion. Therefore, this term is not a peculiarity of the pion-photon TDAs. A similar pion term will be present in the Lorentz decomposition in terms of distribution amplitudes of the axial current for any pair of external particles. A pion exchange contribution has already been analyzed in [20,21] for the axial helicity-flip GPD and, in [16], a similar structure for the axial current has been obtained using different arguments.1This term we have represented
in Fig.2is only nonvanishing in the ERBL region, i.e., the
x 2 ; region. The kinematics of this region allow the emission or absorption of a pion from the initial state, which is described through the PDA. And it can be seen from Fig. 2 that positive values of corresponds to an outcoming virtual pion, whereas negative values of de-scribe an incoming virtual pion. The latter is related to the matrix element hpj qn6 5qj0i, instead of the one
present in Eq. (8), which gives rise to the minus sign included in .
III. A FIELD THEORETIC APPROACH TO THE PION-PHOTON TDA
In Ref. [13] we have defined a method of calculation for the pion GPD in a field theoretical scheme, treating the pion as a bound state of quarks and antiquarks in a fully covariant manner using the Bethe-Salpeter equation. We apply here the same method for evaluating the pion-photon TDAs. This method has enormous advantages because it preserves all the physical invariances of the problem. Therefore, any property as sum rules or polynomiality is preserved.
As usual, we consider that the process is dominated by the handbag diagram. Each TDA has two related contribu-tions, depending on which quark (u or d) of the pion is scattered off by the deep virtual photon. In Fig. 3 we depicted the diagrams in which the photon scatters off the u quark. We observe that there are two kinds of contributing diagrams. In the first one the d antiquark appears as the intermediate state, while in the second the bilocal current couples to a quark-antiquark pair coupled in the pion channel. The latter is present only for the axial current and includes the pion pole contribution.
The details of the method of calculation are given in Ref. [13]. In the present case we obtain, from the first kind of diagram of Fig.3, the following contributions
FIG. 2. Pion pole contribution to the axial bilocal current corresponding to the last term of Eq. (5).
1In order to make this connection it must be realized that there is ap2between our definition of fand the one used in [16], and
Z dz 2 e ixPzhp0; "jqz 2 5q z 2 jpij zz?0 eZ d 4k 24" xP1 2p 0 p 2kTr1 2 1 3 3 iSp0 k 5iSp k k; piSk xP1 22k p p0Tr1 2 1 3 3 iSk k; piSk p 5iSk p0 ; (10)
where Sp is the Feynman propagator of the quark and
k; p is the Bethe-Salpeter amplitude for the pion. Here Tr represents the trace over spinor, color, and flavor indices. The first contribution in Eq. (10) is the one de-picted in the first diagram of Fig.3. The second contribu-tion corresponds to a similar diagram but changing quarks
uand d. In the NJL model,
k; p is as simple as
k; p gqq5p2; (11) where gqq is the pion-quark coupling constant defined in Eq. (A5).
The vector TDA has contribution only from this first kind of diagram. We can express Vx; ; t as the sum of the
active u-quark and the active d-quark distributions. The first contribution will be proportional to the d’s charge, and the second contribution to the u’s charge. Therefore, we can write Vx; ; t 1 3v u!dx; ; t 2 3v d!ux; ; t: (12) Isospin relates these two contributions, v
d!ux; ; t
vu!d x; ; t. A direct calculation gives
vu!d x; ; t 8Ncfgqqm~I3vx; ; t; (13) with ~ I3vx; ; t i Z d4k 24 x 1 k P 1 k2 m2 i p0 k2 m2 i p k2 m2 i: (14)
In this integral, we first perform the integration over k. The pole structure of the integrand fixes two nonvanishing contributions to v
u!dx; ; t, the first one in the region
< x < 1, corresponding to the quark contribution, and
the second in the region < x < , corresponding to a quark-antiquark contribution. Given the relation (12), the support of the entire vectorial TDA, V
x; ; t, is there-fore x 2 1; 1. The analytical expression for (14) is given by Eq. (A11). For the 0, the contributions of the first type of diagram (Fig.3) can be related to v
u!dx; ; t, V0 u Qu 2 p v u!dx; ; t v u!dx; ; t; V0 d Qd 2 p v u!dx; ; t v u!dx; ; t: (15)
Turning our attention to the axial TDA, we find a new contribution arising from the second diagram of Fig. 3. This second contribution comes from the rescattering of a
qqpair in the pion channel. Therefore
Z dz 2 e ixPzhp0; "jqz 2 5q z 2 jpij zz?0 10 X i Mi 2ig 1 2gpstN i; (16)
where i is an isospin index,
FIG. 3. Diagrams contributing to the TDA. We have depicted diagrams in which a quark u is change into a quark d by the bilocal current. There are similar diagrams in which the antiquark dis changed into a u.
Mi eZ d4k 0 24" Tr k0; piSk01 2 1 3 3 iSp0 k0i 5iiSp k0 Tr k0
; piSk0 pi5iiSk0 p01 2 1 3 3 iSk0; (17) Ni Z d 4k 24 xP1 2p 0 p 2k Tr iSp0 k 5iSp ki5i; (18) and psis the pseudoscalar polarization,
ps2 i Z d4k
24fTr i5iSki5iS k: (19) We can now evaluate the axial current in a straightforward way. Nevertheless, in order to extract the axial TDA we must subtract the pion pole contribution. We need for that
the pion amplitude which, in the NJL model, is
x mgqqNc 42f X2 i1 cilog m2 m2 x1 x m2 i m2x1 x ; 0 x 1; (20)
where ciand miare defined in the Appendix. Now, after a
long but direct calculation, we obtain the expression for
Ax; ; t. As the vector TDA, Ax; ; t can be expressed as a sum of the contributions coming from the active u quark and the active dquark. The first one will be proportional to the d’s charge and the second to the u’s charge.
Ax; ; t 1 3a u!dx; ; t 2 3a d!ux; ; t: (21) Isospin relates these two contributions, a
d!ux; ; t a
u!dx; ; t, where the minus sign is originated in
the change in helicity produced by the 5 operator. The expression for a
u!dx; ; t depends on the sign of . In the
jj < x < 1 region we have au!d x; ; t 8Ncfgqqm 1 m 2 x 1 xt 2m2 t1 1 1 ~ I3vx; ; t 1 2m2 t1 1 1 1 162 X 2 i1 ci logm 2m2 i zm2 m2 im2 zm2 ; (22)
with the abbreviations z 1 x x=1 2. And in the jj < x < jj region we have
a u!dx; ; t 8Ncfgqqm 1 m 2 x 1 xt 2m2 t1 1 1 ~ I3vx; ; t 1 1 162 X2 i1 ci 1 2m2 t1 log4 2m2 xtm2 i ym2 42m2 i xtm2 ym2 2 t m2 log4 2m2 xt42m2 i xm2 42m2 i xt42m2 xm2 ; (23)
where x 2 x2 and y z for > 0 and y 0 for < 0.
The expressions for both the vector and axial TDAs in the chiral limit, i.e., m 0, are well defined. In particular, for
t 0, we find the following simple expression for v
u!dx; ; t: v u!dx; ; 0 2 p f6F V 0 2 x2 x jj 2jj1 jj x jj1 x 1 x 1 2 ; (24)
where FV 0 is the chiral limit of the vector pion form factor at zero momentum transfer FV 0 limm!0F
V 0=m 0:17 GeV1.
A similar expression is obtained for a
u!dx; ; t in the chiral limit for t 0
a u!dx; ; 0 2 p f6FA 0 2 x2 x 421 jj x x jj 1 jj x jj1 x 1 xx 1 21 ; (25)
with FA 0 the axial form factor at t 0 in the chiral limit FA 0 limm!0F A 0=m F V 0.
IV. SUM RULES AND POLYNOMIALITY The vector TDA of the must obey the sum rule given in Eq. (6) with the expression of the vector pion form factor in the NJL model given by Eq. (A7). We have numerically recovered the sum rule for different t values. In particular we obtain the value F
V 0 0:0242 for the vector form
factor at t 0, which is in agreement with the experimen-tal value FV0 0:017 0:008 given in [22].
The 0distribution must satisfy the following sum rule [6]: Z1 1 dxQuV0 u x; ; t QdV 0 d x; ; t p2fFt: (26)
A theoretical prediction of the 0 form factor value is given in [23]. In particular, at t 0, the value F0
0:272 GeV1 is found. The neutral pion form factor is directly related to the vector form factor so that the sum rule is satisfied. We obtain the value F
0
0:244 GeV1. In [24], a dipole parametrization based on experimental data is proposed for the t dependence of
F
t, obtaining for the dipole mass 0:77 GeV.
We have found that, for small values of t, the NJL neutral pion form factor can be parametrized in a dipole form with 0:81 GeV.
The axial TDA obeys the sum rules given by Eq. (7) with the axial form factor given in the NJL model by Eq. (A9). This sum rule is satisfied for different t values. The nu-merical results also coincide. In particular we obtain
FA0 0:0239 for the axial form factor at t 0, which is about twice the value F
A 0 0:0115 0:0005 given
by the PDG [22].
We expect TDAs to obey the polynomiality condition. However, no time reversal invariance enforces the poly-nomials to be even in the momenta like for GPDs. That means that the polynomials should be ‘‘complete,’’ i.e.,
they should include all powers in , Z1 1 dxxn1Vx; ; t X n1 i0 Cn;iti: (27)
However, in the chiral limit and for t 0, we have analytically found that the odd powers in go to zero for the polynomial expansion of the vector TDA. A study of the polynomiality in the limit given in Eq. (24) leads to the following analytical expression for the coefficients C n;2it:
C n;2i0 p2f2FV 01 21n11 n 1 n 1 : (28) Notice they do not depend on i. We have numerically tested the polynomiality and obtained it. We observe that the coefficients for the odd powers in are of one order of magnitude smaller than those for the even powers in . In particular we have numerically proved that, in the chiral limit, the polynomials only contain even powers in for any value of t, Z1 1 dxxn1 lim m!0 Vx; ; t X n1=2 i0 C n;2it2i: (29)
The coefficients in the chiral limit, Eq. (29), as well as the coefficients for m 140 MeV, Eq. (27), we numerically obtained are given in TableI.
The dependence of the moments of the axial TDA
Ax; ; t also has a polynomial form. Those polynomials contain all the powers in , i.e., even and odd,
Z1 1dxx n1Ax; ; t X n1 i0 C0n;iti: (30)
An analytic study of the polynomiality in the limit given in Eq. (25) confirms that all the powers in have to be present. The analytic values for the coefficients in this specific limit are
TABLE I. Coefficients of the polynomial expansion for the vector TDA. The pion mass is expressed in MeV and t is expressed in GeV2. Notice that the coefficients have to be multiplied by 103.
1 2 3 4 n C1;0t C2;0t C2;1t C3;0t C3;1t C3;2t C4;0t C4;1t C4;2t C4;3t m 0 t 0 22.6 22:6 0.0 3.77 0.0 3.77 6:7 0.0 6:7 0.0 t 0:5 13.7 16:3 0.0 3.00 0.0 2.43 5:7 0.0 4:9 0.0 t 1:0 10.4 13:4 0.0 2.57 0.0 1.90 5:0 0.0 4:0 0.0 m 140 t 0 22.7 22:9 0.44 3.80 0:10 3:67 6:8 0.19 6:8 0.12 t 0:5 13.7 16:4 0.25 3.00 0:06 2.43 5:7 0.13 4:9 0.08 t 1:0 10.3 13:4 0.19 2.57 0:05 1.87 5:0 0.11 4:0 0.06
C0 n;2i0 p2f2F A 01 21n1 n 2i nn 1n 2; C0 n;2i10 p2f2F A 01 21n1 2i 1 nn 1n 2; (31)
which is in agreement with those numerically obtained. The coefficients of the polynomial expansions in the chiral limit are very close to those obtained for the physical values of the pion mass. In Table II, the coefficients
C0n;it are therefore shown only for m 140 MeV. The
polynomiality property of the term containing the PDA can also be studied. The t dependence only comes from the pion pole. We can therefore write a general relation about the polynomiality property of the whole axial bilocal ma-trix element Z1 1 dxxn1Z dz 2e ixpzhP0j qz 2 n5q z 2 jPijzz?0 1 pe" 1 2 p f Xn i0 C00n;iti: (32) V. DISCUSSION
In Figs.4and5, the vector and axial TDAs are plotted in function of x for several values of and t. In these figures, the u quark contributes to the TDAs in the region of x going from jj to 1 and the d antiquark going from 1 to jj. Therefore, for x 2 jj; 1 (x 2 1; jj) only
u-valence quarks (d-valence antiquarks) are present (DGLAP regions). Besides, TDAs in the jj < x < jj region (ERBL region) receive contributions from both types of quarks. For the vector TDA, we observe in Fig. 4 that the position of the maxima is given by the value, separating explicitly the ERBL region from the DGLAP regions. The vector TDA is positive (negative) for negative (positive) values of x with the change of sign occurring around x 0. This change in the sign of the vector TDA is originated in the presence of the electric charge of the quarks in Eq. (12).
The process involved in the calculation of the TDAs allows negative values of the skewness variable. In the chiral limit, the skewness variable goes from 1 to
1, for any value of t. In the chiral limit, the vector
TDA for a negative value of is equal to the vector TDA for jj. This can be seen from the polynomiality expansion in this limit, Eq. (29), which has only even powers of . For the physical value of the pion mass, negative values of
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08 -1.00 -0.50 0.00 0.50 1.00
V(x,
ξ,t)
x
mπ=140 MeV and t=0 GeV2
ξ=0.25 ξ=0.5 ξ=0.75 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 -1.00 -0.50 0.00 0.50 1.00
V(x,
ξ,t)
x
mπ=140 MeV and t=-0.5 GeV2
ξ=0.25 ξ=0.5 ξ=0.75
FIG. 4 (color online). The vector TDA in the case m 140 MeV and, respectively, t 0 and t 0:5 GeV2.
TABLE II. Coefficients of the polynomial expansion for the axial TDA. The pion mass is expressed in MeV and t is expressed in GeV2. Notice that the coefficients have to be multiplied by 103. 1 2 3 n C01;0t C02;0t C02;1t C03;0t C03;1t C03;2t m 140 t 0 22.4 3:77 4.00 6.8 4:9 2.3 t 0:5 16.1 2:97 2.57 5.7 3:5 1.7 t 1:0 13.2 2:53 1.97 5.0 2:8 1.3
are bounded by t=2m2
t < . For each allowed value
of , we found that the numerical value of Vx; jj; t is close to Vx; jj; t, due to the smallness of the coefficients of the odd powers of in the polynomial expansion (27).
Analyzing the axial TDAs, plotted in Fig.5, we observe two different behaviors depending on the sign of . For positive , the position of the minima is given by the value of the skewness variable while the position of the maxima is always x ’ 0 in the ERBL region and x ’ 1 =2 in the DGLAP regions. For negative , the value of the axial TDA at x is important and in some cases is a maximum and, in the ERBL region, Ax; ; t presents a minimum near x 0. As we have previously shown, the axial TDA in the ERBL region receives contributions from two different diagrams, depicted in Fig.3. In the second of these diagrams, a virtual quark-antiquark interacting pair in the pion channel appears. The pion pole, contained in
this diagram, has been subtracted but the remaining non-resonant part contributes to the axial TDA. We observe that the latter contribution is the dominant one and produces the maxima around x 0 for positives (Fig. 6). Now, the axial TDA does not change the sign when we go from negative to positive values of x. In the axial TDA the change of sign originated in the presence of the electric charge of the quarks in Eq. (21) is compensated by the change of sign between quark and antiquark contributions generated by the 5 operator present in the axial current.
By comparing the plots for different values of the mo-mentum transfer, it is observed that the amplitudes are lower for higher ( t) values, as it can be inferred from the decreasing of the form factors with ( t), connected to the TDAs through the sum rules. By increasing the ( t) value, not only the width and the curvature of the TDAs are changed, but we also observe that higher values of are preferred, i.e., the sign of the derivative of the collection of maxima changes passing from a zero momentum transfer to a nonzero one.
Isospin relates the value of the vector and axial TDAs in the DGLAP regions,
Vx; ; t 1
2Vx; ; t; jj < x < 1;
Ax; ; t 1
2Ax; ; t; jj < x < 1;
(33)
being the factor 1=2 the ratio between the charge of the u and d quarks. We observe in Figs.4and5that our TDAs
. . . . . . . . . . . . . .
FIG. 6 (color online). Contributions to the axial TDA for both positive ( 0:25, solid line) and negative ( 0:5, dashed line) values of the skewness variable and for m 140 MeV
and t 0:5 GeV2. In each case, and in the ERBL region, the contribution coming from the first diagram of Fig. 3 is repre-sented by the dash-dotted lines and the nonresonant part of the second diagram of Fig.3is represented by the dotted lines. . . . . . . . . . . . . . . . . . . . . .
FIG. 5 (color online). The axial TDA in the case m
140 MeV and, respectively, t 0 and t 0:5 GeV2.
satisfy these relations. It must be realized that the relation (33) cannot be changed by evolution.
Regarding the chiral limit, we observe that both the vector and axial TDAs do not significantly change going from a nonzero pion mass to the physical mass, except for the change in the lower bound of .
Previous studies of pion-photon TDAs have already been done [16,17]. In both works, double distributions have been used. Even if our model is different, a compari-son of the results is still possible. The aim of the author of the first paper [16] is to provide some estimates of the vector and axial TDAs on the basis of the positivity bounds. The order of magnitude of the obtained amplitudes are similar to ours, but the former are constrained by the FV and FAform factors, through the sum rules. The sum rules are an input imposed in Ref. [16] while it is a result in our calculation. The vector TDA obtained in this paper has some similarities with ours. Nevertheless, it does not sat-isfy the isospin relation (33) due to the different choice of the u-quark and d-antiquark distributions, the first one related to the pion and the second to the photon, used in the saturation of the positivity bounds. We have also studied the positivity bounds for GPDs and noticed that it is actually an upper bound that is sometimes very much higher than the value of the GPD itself. The vector TDA obtained in [16] is rather peaked at x , whereas we have a smoother behavior.
A detailed comparison with the results of Ref. [17] is not easy, due to the choice of the asymmetric notation. Nevertheless the vector TDA seems to be consistent with our results and some limits, in particular, when m 0 and t 0, can be recovered going from one notation to the other.
Regarding our axial TDA, it differs from the two pre-vious calculations [16,17] due to the effect of the non-resonant part of the second diagram of Fig. 3. This contribution, corresponding to the last term of Eq. (23), is proportional to t m2
1 but with zero value for the
residue. The presence of this term is crucial in order to obtain the axial form factor using the sum rule. In fact, its contribution in Eq. (A9) can be easily recognized. Furthermore this term is dominant in the ERBL region as we can infer from Fig.6.
VI. CONCLUSIONS.
In this paper we have defined the pion-photon vector and axial transition distribution amplitudes using the Bethe-Salpeter amplitude for the pion. In order to make numerical predictions we have used the Nambu–Jona-Lasinio model. The Pauli-Villars regularization procedure is applied in order to preserve gauge invariance.
We know from PCAC that the axial current couples to the pion. Therefore, in order to properly define the axial TDA (all the structure of the incoming hadron being in-cluded in Ax; ; t), we need to extract the pion pole
contribution. In so doing, we found that the axial TDA had two different contributions, the first one related to a direct coupling of the axial current to a quark of the incoming pion and to a quark coupled to the outcoming photon and a second related to the nonresonant part of a quark-antiquark pair coupled with the quantum numbers of the pion.
The use of a fully covariant and gauge invariant ap-proach guarantees that we will recover all fundamental properties of the TDAs. In this way, we have the right support, x 2 1; 1, and the sum rules and the polyno-miality expansions are recovered. We want to stress that these three properties are not inputs, but results in our calculation. The value we found for the vector form factor in the NJL model is in agreement with the experimental result [22], whereas the value found for the axial form factor is two times larger than in [22]. This discrepancy is a common feature of quark models [17]. Also the neutral pion vector form factor F
t is well described. These
results allow us to assume that the NJL model gives a reasonable description of the physics of those processes at this energy regime.
Turning our attention to the polynomial expansion of the TDAs, we have seen that, in the chiral limit, only the coefficients of even powers in were non-null for the vector TDA. No constraint is obtained for the axial TDA. Nevertheless, the NJL model provides simple expressions for the coefficients of the polynomial expansions in the chiral limit, Eqs. (28) and (31).
We have obtained quite different shapes for the vector and axial TDAs. This is in part, at least for the DGLAP regions, imposed by the isopin relation (33). We have pointed out the importance of the nonresonant part of the
qqinteracting pair diagram for the axial TDA in the ERBL region.
It is interesting to inquire about the domain of validity of the relations (33). These relations are obtained from the isospin trace calculation involved in the central diagram of Fig. 3. Because of the simplicity of the isospin wave function of the pion these results are more general than the NJL model and could be considered as a result of the diagrams under consideration. These diagrams are the simplest contribution of handbag type.
The transition distribution amplitudes proposed by the authors of [6] open the possibility of enlarging the present knowledge of hadron structure for they generalize the concept of GPDs for nondiagonal transitions. Calculated here, as a first step, for pion-to-photon transitions, these new observables should lead to interesting estimates of cross section for exclusive meson pair production in
scattering [7].
ACKNOWLEDGMENTS
We are thankful to J. P. Lansberg, B. Pire, L. Szymanowsky, and V. Vento for useful discussions.
This work was supported by the sixth framework program of the European Commission under Contract No. 506078 (I3 Hadron Physics), MEC (Spain) under Contracts No. BFM2001-3563-C02-01 and No. FPA2004-05616-C02-01 and Grant No. AP2005-5331, and Generalitat Valenciana under Contract No. GRUPOS03/094.
APPENDIX
In Sec. II we have introduced the minimal kinematics definitions. The explicit expressions for the pion and pho-ton momenta in terms of the light-front vectors are
p 1 p1 2 ?1 2 P21 1 2m 2 n; (A1) p0 1 p1 2 ?1 2 P21 1 2m 2 n: (A2) Here we have P2 p p0=22 m2 =2 t=4, p0 p, ? 0; 1; 2; 0 0; ~?; 0, and 2 t. The polarization vector of the real photon, ", must satisfy the transverse condition, " p0 0, and an additional gauge fixing condition. When deriving Eq. (5), we need "
n "=Pto kinematically become higher twist, i.e., "
n ! 0 when P! 1. The standard gauge fixing condi-tions, "0 0 or " 0, satisfy the previous requirement. In fact, all that we need is that the components of the polarization vector remain finite when P goes to infinity. In Sec. III we use the NJL model. We follow the notation of Ref. [13] and we refer the reader to this paper for more details. The NJL model considers the Lagrangian
L qxi@6 0qx g qq2 qi5q~ 2; (A3) where 0 is the current quark mass. As it is well known, the first consequence of the scalar interacting term is to provide a constituent quark mass, m, different from the current mass. Because of the pointlike character of the
interaction, this Lagrangian is not renormalizable. We shall use the Pauli-Villars regularization in order to render the occurring integrals finite. This means that for integrals like the ones defined in Eq. (14), we make the replacement
Z d4p 24fp; m 2 !Z d 4p 24 X2 j0 cjfp; m2j; (A4) with m2 j m2 j2, c0 c2 1, c1 2. Following Ref. [8] the regularization parameters and m are deter-mined by fitting the pion decay constant and the quark condensate (in the chiral limit). With the conventional values h uui 250 MeV3 and f
93 MeV, we get
m 241 MeV and 859 MeV.
The pion-quark coupling constant is given by
g2qq 1 12I2m2 m2@I2p=@p2p2m2 ; (A5) with I2p2 i Z d4k 24 1 k2 m2 ik p2 m2 i 1 162 X2 j0 cj logm 2 j m2 2 4m2 j p2 1 v u u t arctan 1 4m2 j=p2 1 q : (A6)
The numerical value of the pion-quark coupling constant is
g2
qq 6:36 for the physical value of the mass of the
pion, and g2
qq 6:71 in the chiral limit.
In the NJL model, the vector form factor is
FVt 8Nc 3 mm gqq 2 p I3p; p0: (A7)
In order to calculate the form factors we need the expres-sion for the three-propagator integral. In the particular case where p2 m2
and p02 0, the expression for I3p; p0 is
I3p; p0 iZ d4k 24
1
k2 m2 ip k2 m2 ip0 k2 m2 i
1 42 X2 j0 cj Z1 0 dx x p logx 2t x1 xm2 2m2j p x2t x1 xm2 2m2j p ; (A8) with t2x4 x21 x2m4 2tx2m2x1 x 2m2j, t p 0 p2.
The axial form factor involves the three-propagator integral as well, but also the two-propagator integral given by Eq. (A6) F A t 4Ncmmgqq 2 p I3p; p0 2 m2 t I2m2 I2t : (A9) The 0 !
form factor can be obtained from the vector form factor through an isospin rotation: Ft
2 p
F
V t=m. In the chiral limit and for t 0 all the considered form factors have the simple expression:
F 0 2 p m!0 F V 0 m m!0 F A 0 m m!0 1 2 p 42f 1 2m 2 m2 2 m2 m2 22
0:192 1 0:108 GeV1 0:171 GeV1: (A10)
The first coefficient of the right-hand side is what is expected from the axial anomaly contribution to ! decay. The term between brackets has a small correction to the expected value of 1 due to the finiteness of the regularization masses. In the NJL model not only the quarks, but also the counterterms, run in the triangle diagram of the axial anomaly. In a proper renormalizable theory this correction disappears in the limit ! 1.
In the light-front calculations we need two kind of integrals. The first one is
~ I2x; ; t i Z d4k 24 x 1 k=P k p12 m2 i k p 22 m2 i b xx a 1 42 1 a b X2 j1 cjlog m2b a2 b xx ap 2 p12 m2 jb a2 b xx ap2 p12 ; (A11)
with b max1 p1=P; 1 p2=P and a min1 p
1=P; 1 p2=P. The second light-front integral needed in our calculations is the one defined by Eq. (14). This integral can be performed by standard methods, obtaining
~ I3vx; ; t 1 322 X2 j0 cj 1 D p logt=21 x C D p t=21 x C p ;D (A12) with D m2 2 x t 21 x 2 m2 j1 2m2 t1 ; C 8 > > > < > > > : m2 2 x m2j 12 1x ; jj < x < 1; m2 2 x m2j 21 x ; jj < x < jj > 0; m2 2 x m2j 21 x ; jj < x < jj < 0:
As an application of the expression (A11) we can evaluate the PDA. From its definition in Eq. (8) we have
x 4Ncmgqq f iZ d 4k 24 x 1 k=p k2 m2 i k p2 m2 i: (A13) This integral is of the type of (A11) with a 0 and b 1. Therefore,
x Ncmgqq 42f x1 xX 2 j1 cjlogm 2 m2 1 xx m2 j m21 xx : (A14)
As it is expected for the PDA, x runs from 0 to 1. We can test our result verifying that in the chiral limit x
x1 x, as it is well known for the NJL model. This peculiarity of the NJL model in this limit is present also in the
quark valence distribution, which becomes as simple as qx x1 x. In Ref. [14] is discussed how this is, in fact, a quite reasonable approximation of realistic models.
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